# Central factor satisfies intermediate subgroup condition

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., central factor) satisfying a subgroup metaproperty (i.e., intermediate subgroup condition)

View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties

Get more facts about central factor |Get facts that use property satisfaction of central factor | Get facts that use property satisfaction of central factor|Get more facts about intermediate subgroup condition

## Contents

## Statement

### Verbal statement

A central factor of a group is also a central factor in every intermediate subgroup.

## Related facts

### Generalizations and other particular cases of the generalizations

- Left-inner implies intermediate subgroup condition
- Left-extensibility-stable implies intermediate subgroup condition
- Normality satisfies intermediate subgroup condition
- Centrality satisfies intermediate subgroup condition
- Cocentrality satisfies intermediate subgroup condition

### Related metaproperties for central factor

- Central factor does not satisfy transfer condition
- Central factor satisfies image condition
- Central factor is upper join-closed

## Proof

### Using function restriction expressions

This subgroup property implication can be proved by using function restriction expressions for the subgroup properties

View other implications proved this way |read a survey article on the topic

*This proof method generalizes to the following results:* left-inner implies intermediate subgroup condition, left-extensibility-stable implies intermediate subgroup condition

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### Proof using product with centralizer definition

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